Question:

Is the belief in "days" lasting for potentially long periods of time an attempt to rationalize Creation with an Old Earth or with Evolution?

Response: 

Amongst Christians, the origin of the idea that a day could be some long period of time is mostly credited to 2 Peter 3:8, where Peter explains that to the Lord, "a thousand years is like a day" (and vice versa).  Consequently, certain scholars have applied this text (and others) towards interpreting the days of creation as long periods of time.  I believe one of the first examples of such exegesis is Augustine who rejected the idea that God created the earth in six 24 hour days, based on his own exegesis of the Genesis text.  What is remarkable is that Augustine's view was formulated a thousand years before the discovery of fossils and various scientific theories of evolution and cosmic beginnings.  

Anyway, the Old Earth View of "day" is popular amongst evangelicals because it does allow for a synthesis of scientific theory with Genesis 1-2.  Within the Evangelical movement, I think you will find God-fearing Christians who do and do not embrace Evolution, but generally, all Christians reject the notion that Humanity evolved.  As for the evolution of animals, fish, and plant life, you may find varied opinions.  I would say the Bible is silent on the particulars of creating fauna and flora.  What is clear is that Humanity is God's primary concern, both in the Creation narrative and beyond.  Other life forms are not near as important.

In Christ, Marty